Deformation force constants

Stretching force constants in mdyn/A and deformation force constants in mdyn A/radian. ... 

In some cases (such as torsional barrier terms) it is possible to do this definitively, while in others (such as valence angle deformation force constants and ideal distances and angles) it is not. However, useful starting points for the empirical refinement can be derived from experiment. 

This is a consequence of variation of the mass of X, the C-X stretching force constant, and the X-C-H deformation force constant. Table 4.1-5 shows the variation of this vibration with the position of the X atom within the periodic system. Colthup (1980) describes model calculations by the CNDO/2 method which explain this behavior. 

The assignments and subsequent calculations of Pt(CN)4 (4) and of Pd(CN)4 (6) are not consistent with the previous ones. In particular, the deformation force constants are abnormally weak. Thus, we propose a new assignment in analogy to Ni(CN)4 and Au(CN)4, but some caution must be observed in a frequency range where internal deformations and external lattice vibrations can be confused. 

For every type of angle including three atoms, two parameters (force constant fe and reference value 0q) are needed. Also, as in the bond deformation case, higher-order contributions such as that given by Eq. (23) are necessary to increase accuracy or to account for larger deformations, which no longer follow a simple harmonic potential. 

If atom i or atom j is a hydrogen, the deformation (r-r ) is considered to be zero. Thus, no stretch-bend interaction is defined for XH2 groups. The stretch-bend force constants are incorporated... 

This equation shows that at small deformations individual chains obey Hooke s law with the force constant kj = 3kT/nlo. This result may be derived directly from random flight statistics without considering a network. 

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

Fig. 6. The three basic curvature deformations of a nematic Hquid crystal bend, twist, and splay. The force constants opposing each of these strains are...

The deformation of soft surfaces can be minimized with SFM by selecting cantilevers having a low force constant or by operating in an aqueous environment. The latter eliminates the viscous force that arises from the thin film of water that coats most surfaces in ambient environments. This viscous force is a large contributor to the total force on the tip. Its elimination means that the operating force in liquid can be reduced to the order of 10 N. 

According to the transition state theory, the pre-exponential factor A is related to the frequency at which the reactants arrange into an adequate configuration for reaction to occur. For an homolytic bond scission, A is the vibrational frequency of the reacting bond along the reaction coordinates, which is of the order of 1013 to 1014 s 1. In reaction theory, this frequency is diffusion dependent, and therefore, should be inversely proportional to the medium viscosity. Also, since the applied stress deforms the valence geometry and changes the force constants, it is expected... 

A detailed discussion about the functional form for f(v[r) can be found in Ref. [15]. The frequencies of molecular vibrations depend on the force constants which are themselves attributed to the bond geometry. It is then not surprising that useful information on bond deformation under stress can come from IR or Raman spectroscopy. 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

According to Eq. (11), the force constant for the normal vibration Q, can be identified with the term in braces and can be negative if the second term, which is positive, exceeds the first term. If the force constant is negative, the energy should be lowered by the nuclear deformation Qi, and the second-order distortion from the symmetrical nuclear arrangement would occur spontaneously. 

In order to understand the mechanical properties of polymers it is useful to think of them in terms of their viscoelastic nature. Conceptually we can consider a polymeric item as a collection of viscous and elastic sub-components. When a deforming force is applied, the elastic elements deform reversibly, while the viscous elements flow. The balance between the number and arrangement of the different components and their physical constants controls the overall properties. We can exploit these relationships to create materials with a broad array of mechanical properties, as illustrated briefly by the following examples. 

The summations in Eq. (8) and (9) usually extend over all internal parameters, independent and dependent, i.e. the potential constants in these expressions are also not all independent. For example, the nonsymmetric tetrasubstituted methane CRXR2R3R4 possesses five independent force constants for angle deformations at the central carbon atom, whereas in our calculations we sum over the potential energy contributions of the six different angles (only five are independent ) at this atom using six different potential constants for angle deformations. The calculation of the independent force constants, which is necessary for the evaluation of the vibrational frequencies, will be dealt with in Section 2.3. 

Terms representing these interactions essentially make up the difference between the traditional force fields of vibrational spectroscopy and those described here. They are therefore responsible for the fact that in many cases spectroscopic force constants cannot be transferred to the calculation of geometries and enthalpies (Section 2.3.). As an example, angle deformation potential constants derived for force fields which involve nonbonded interactions often deviate considerably from the respective spectroscopic constants (7, 7 9, 21, 22). Nonbonded interactions strongly influence molecular geometries, vibrational frequencies, and enthalpies. They are a decisive factor for the transferability of force fields between systems of different strain (Section 2.3.). 

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